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Chapter 1: Q. 44 (page 97)

For each limit limx鈫�cf(x)=Lin Exercises 43鈥�54, use graphs and algebra to approximate the largest value of 未such that if x鈭�(c-未,c)鈭�(c,c+未)thenf(x)鈭�(L-蔚,L+蔚).

limx鈫�2x3=8,蔚=0.25

Short Answer

Expert verified

The largest value of未鈮�0.0206.

Step by step solution

01

Step 1. Given Information.

The given expression islimx鈫�2x3=8,蔚=0.25.

02

Step 2. Explanation. 

From the given expression, we have, c=2,L=8.

The limit expression can be written as a formal statement as below,

For all epsilon positive, there exists a delta positive such that if
x鈭�(2-未,2)鈭�(2,2+未)Thenx3鈭�(8-蔚,8+蔚)

Now, the largest value of delta is given by,

x3=L+蔚x3=8+0.25x3=8.25x=8.2513

Thus,

未=8.2513-2未鈮�0.0206

03

Step 3. Conclusion.

The largest value of未鈮�0.0206.

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Most popular questions from this chapter

Calculate each of the limits:

limx鈫�3xtan-1x

Sketch a labeled graph of a function that satisfies the hypothesis of the Intermediate Value Theorem, and illustrate on your graph that the conclusion of the Intermediate Value Theorem follows.

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point x=c.

f(x)=2鈭�x,鈥呪赌呪赌呪赌�ifxrationalx2,鈥呪赌呪赌呪赌�ifxirrational,c=1

Explain why the Intermediate Value Theorem allows us to say that a function can change sign only at discontinuities and zeroes.

Use algebra to find the largest possible value of 未 or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.

Ifx>N, then1x2鈭�0<0.001
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